Contracts similar to options are believed to have been used since ancient times.

In London, puts and "refusals" (calls) first became well-known trading instruments in the 1690s during the reign of William and Mary.

Privileges were options sold over the counter in nineteenth century America, with both puts and calls on shares offered by specialized dealers.

But how had traders actually priced the simplest options before the Black–Merton-Scholes model became common knowledge? What were the most popular methods to determine arbitrage-free prices before 1973?

8 Answers
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You may want to look at Chapter 5 - "The Quest for the Option Formula" from the Derivatives. The book is available online for free and it has a very descent review of approaches that were used 20-30 years before the Black-Scholes-Merton equation.

The man who grasps principles can successfully select his own methods. The man who tries methods, ignoring principles, is sure to have trouble.

~ Ralph Waldo Emerson ~

Black-Scholes made it possible for an idiot with a calculator to imagine that he was smart enough to judge the value of options ... it has always been possible to determine option value -- Black-Scholes is not necessarily the best approach.

In his preface to the sixth edition of Security Analysis, noted value investor Seth Klarman discusses how Graham and Dodd's methods mentioned in the original edition (copyright 1934) are more relevant than ever. Specifically, Mr. Klarman suggests how he has successfully applied those methods. Mr. Klarman is relatively-famous for his use of option in an investment strategy has been been significantly more profitable than average. Mr. Klarman has taken advantage of less well-informed investors who falsely pride themselves on their mathematical sophistication and familiarity with formulas like Black Scholes. Those investors "plug and chug" their numbers into their quantitative models [without having a clue about the fundamentals driving the probability distribution upon which the mathematics of the formula are based], receive their below average returns and never realize that the problem when almost everyone is using Black-Scholes, the market is ripe for a successful contrarian anti-Black-Scholes strategy.

An option pricing strategy based upon value investing precepts would be driven by a rigorous analysis of downside potential and a similar analysis of upside potential. The rigor necessary for understanding the sensitivity of the stock to various scenarios would inform the judgement necessary to determine the likelihood of those different scenarios. Option pricing based upon Graham and Dodd's methods would not be driven by irrationally sanguine market-driven estimates of either implied or historical volatility ... it would reflect actual risks faced by the underlying assets, not the market assessment of those risks ... as you will recall, value investing is based upon the premise that the market-based assessments are frequently very, very wrong.

Anyone seeking additional insights into Seth Klarman's methodoloy, should obtain a copy of Security Analysis and a copy of Mr. Klarman's own text Margin of Safety. Of course, there are plenty of other reasons to read, re-read, re-re-read Security Analysis and Margin of Safety beyond just an alternative to Black-Scholes.

I think this slightly misses the point. Before Black-Scholes options prices were set entirely by human judgement, just like prices in many other markets are set, which is why this model was so important. Peter Bernstein has a good recollection of this kind of behavior in "Capital Ideas".

There is a missing link to early options pricing literature which had been overlooked. Put-call parity along with static delta hedging were understood in actionable detail well before BSM and trading and risk management were accomplished through heuristic methods which indeed continued to be used after BSM.

Options and futures were common instruments in France at the end of the 19th century. Louis Bachelier, in his 1900 thesis, derives the price of a European option when the underlying asset is normally distributed. Interestingly, he seems to have some strong opinions about mathematical finance in his introduction to his thesis:

The calculus of probabilities will undoubtedly never be applied to the movement of stock prices, and the dynamics of the stockmarket will never be an exact science.

To add on to what others have said: the formula still does not provide a price -- just a way to calculate "implied" volatility. The BSM calculates a hypothetical value (using binary branchings as the storytelling tool) and this hypothetical merely provides a reference for this common "what-if" question.

The only sense in which "arbitrage free" entered the minds of traders before BSM was "Am I being cheated?"

One more thing to add: the volume of derivatives trading exploded after the BSM formula, so there wasn't that much derivatives trading going on beforehand.

Another source is Derivative Pricing 60 Years before Black–Scholes: Evidence from the Johannesburg Stock Exchange by LYNDON MOORE and STEVE JUH

From the abstract:

We obtain daily data for warrants traded on the Johannesburg Stock
Exchange between 1909 and 1922, and for a broker’s call option quotes
on stocks from 1908 to 1911. We use this new data set to test how
close derivative prices are to Black–Scholes (1973) prices and to
compute profits for investors using a simple trading rule for call
options. We examine whether investors exercised warrants optimally and
how they reacted to extensions of the warrants’ durations. We show
that long before the development of the formal theory, investors had
an intuitive grasp of the determinants of derivative pricing.